Tagged Questions

Questions on (ordinary) differential equations. For questions specifically concerning partial differential equations, use the (pde) tag.

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Calculate half life of esters

I'm trying to calculate the level of testosterone released from different testosterone esters. Here are some graphs of testosterone levels after single injections of 250mg of each ester. Testo U ...
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Procedure for 3 by 3 Non homogenous Linear systems (Differential Equations)

Here is the problem I have. $$x^{'}(t)=Ax(t)+f(t)$$ where $A=\begin{pmatrix} 5&-3&-2\\8&-5&-4\\-4&3&3 \end{pmatrix} f(t)=\begin{pmatrix} -\sin (t)\\ 0 \\ 2 \end{pmatrix}$ I am ...
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Is the continuity of a vector field enough for the existence of the solution of a differential equation?

I've recently seen the existence-uniqueness theorem for ordinary differential equations from Arnold's book. I understand that the theorem as stated guarantees both existence and uniqueness if the ...
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Differential operators confussion

I want to solve this PDE: $$u_t-6uu_x+u_{xxx} = 0\,(1)$$ with the Inverse Scattering Method. This method is based on showing that the above equation can be expressed as $$L_t=LB-BL,\,(2)$$ where $L$ ...
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How can I prove that the DE $y'=y^\alpha$ has infinitely many solutions?

I need to show that the DE $y'=y^{\alpha}$, where $\alpha$ is a constant with $0<\alpha<1$, has infinitely many solutions passing through the point $(0,0)$. Also I need four of such solutions. ...
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Choice of the First Term in Legendre Polynomials

The two solutions of the Legendre's Differential Equation obtained by series solution method are : and Now according to my textbook, for the useful polynomial for n equal to a positive integer, ...
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Compute $\int_cd\omega$ and $\int_{\partial c}\omega$

Question: Let $c:I^2\rightarrow\mathbb{R}^3$ be the singular $2$-cube given by $$c(s,t)=\left(\frac{1}{2}s^2,st,\frac{1}{2}t^2\right)$$Let $x=(x,y,z)$ denote the cartesian coordinates on $\mathbb{R}^3$...
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Numerical estimates for the convergence order of trapezoidal-like Runge-Kutta methods

I have to calculate approximations of the solution with the method $$y^{n+1}=y^n+h \cdot [\rho \cdot f(t^n,y^n)+(1-\rho) \cdot f(t^{n+1},y^{n+1})] ,\quad n=0,\ldots,N-1 \\ y^0=y_0$$ for various ...
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Generalized “Worm on the rubber band ” problem

I found this « Worm on the rubber band » problem in Concrete Mathematics book. A slow worm $W$ starts at one end of a meter-long rubber band and crawls one centimetre per minute toward the other end. ...
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How to solve following system of ordinary differential equations using Octave? $$\frac{dx}{dt} = - [x(t)]^2 - x(t)y(t)$$ $$\frac{dy}{dt} = - [y(t)]^2 - x(t)y(t)$$ Update: initial conditions: $x(t=... 2answers 50 views Differentiation - simple case In the book calculus made easy, page 22 the case of the negative power for$y=x^{-2}\begin{align} y+dy & =(x+dx)^{-2}\tag{1}\\ \\ & = x^{-2}\left(1+\frac{dx}{x}\right)^{-2}\tag{2} \... 3answers 370 views Tricky Separable Differential Equation Please guide me: y' + ay +b = 0 (a not zero) is supposed to be separable and has solution y = ce^{-ax} - \frac ba Here is my start to this problem: \frac{dy}{dx} + ay = -b is as far as I ... 0answers 14 views IVP Using Numerical Methods Suppose that y(t) is the exact solution of the ivpy'(t)=f(t,y(t)), y(0)=y_0$$and u(t) is any approximation to y(t) with u(0)=y(0). Define the error e(t)=y(t)-u(t). How can I show that ... 3answers 106 views Solve y'-\int_0^xy(t)dt=2 I have not idea how to approach this differential equation.$$y'-\int_0^xy(t)dt=2$$. Basically, I did,$$F''(t)-F(x)+F(0)=2 \;\;\;\;\;\;\; F'=y$$I am stuck. Thank You. 1answer 85 views Solutions and attraction regions of following odes? Assume a mapping X: \mathbb{R} \to \mathbb{R}^d. We know that the solution to ode$$ d X_t = (\mu - X_t) dt $$is X_t = (X_0-\mu) e^{- t} + \mu, which indicates that X_t converges to \mu as ... 1answer 46 views general conditions for reverse poincare inequality I'd like to know when the reverse Poincare inequality is true: Given a bounded domain \Omega, and f \in L^2(\Omega), under what conditions on f (neglecting the trivial constant case) and/or \... 2answers 5k views Difference between improper node and proper node for phase portrait Can someone offer an explanation for the difference between these two? I see pictures of what seem to be examples of both, but it's hard for me to discern what a new portrait would be. Any help? ... 3answers 456 views General Solution for a given system of equations Find the general solution of this system of equations:$$x' = \pmatrix{-1&0&0\\1&0&-1\\1&1&0}x$$I got the eigenvalues to be: \lambda = -1,\pm i The eigenvectors ... 3answers 573 views Is f'(x)-3f(x) = 0 subspace of differentiable functions f\colon (0,1)\to \mathbb{R} V is space of differentiable functions f(0,1) \to \mathbb{R} and W is a subset of f that meets f'(x) - 3f(x) = 0 for all x\in (0,1). Is subset W a subspace of V? I know that I have ... 1answer 47 views H0w are second order nonlinear ordinary differential equations solved? I conceived the following second order nonlinear ordinary differential equation:$$\frac{d^2y(x)}{dx^2}=\frac{k}{(y(x))^2}$$I can tell it's nonlinear because of the \frac{k}{(y(x))^2} term and ... 1answer 81 views Initial Value Problem Please help solving the following initial value problem:$$y''-3y'+2y \; = \; 3e^{-x}-10 \cos {3x}; \;\;\; y(0)= 1, \;\;\; y'(0)=2 $$I have been working at it and have been hitting a road block 2answers 69 views trouble with non-homogeneous ODE system… which method shall I use? I am an undergrad statistics student and I am having troubles with non-homogeneous ODE systems. During my classes I went over just three methods for solving odes: Laplace transform, Fourier transform ... 1answer 817 views Finding the interval where a solution is certain to exist for the equation y' + (\tan t)y = \sin t Given the following problem: Determine (without solving the problem) an interval in which the solution is certain to exist for the initial value problem y' + (\tan t)y = \sin t, \space y(2\pi) = ... 3answers 336 views Differential Equations Skydiver Problem I've seen many variants of this problem online, but not quite the same as this, so I don't believe this is a duplicate. The famous differential equation problem models a skydiver jumping out of a ... 1answer 137 views How many f(x) are possible satisfying f(x)=f'(x) and f(0)=f(1)=0. Let f:[0,1]\to\Bbb{R} be a fixed continuous function such that f is differentiable on (0,1) and f(0)=f(1)=0. Then the equation f(x)=f'(x) admits how many solutions? The only solution that ... 2answers 34 views Find the first order system of linear equations Regard the diff equation: mϕ′′+aϕ′+(mg/L)ϕ=0 ϕ(0)=0.1 ϕ′(0)=0 where m=0.1,L=1,a=2, 1) Rewrite the second order diff equation as a system of first order linear equations. 2) What is the ... 6answers 155 views Solution to a second order ordinary differential equation Let \beta >0, \gamma > 0, \omega > 0 and \xi >\xi_0 . The question is to show that the solution to the following inhomogenous ordinary differential equation: \... 1answer 128 views Find the velocity of a flow The question is: Find the velocity of the flow described by the velocity potential given in the polar coordinates φ$$(r, θ) = $θ$, where $x = r cos θ$ and $y = r sin θ$, $r > 0, 0 ≤ θ < 2π$ ...
Consider the system $$\dot{x}=y,\quad\dot{y}=z,\quad\dot{z}=-x-y-z+x^2+az^3.$$ Check that the zero equilibrium has a pair of pure imaginary eigenvalues. Make a linear coordinate ...